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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Solving sparse finite element problems on neuromorphic hardware

The finite element method (FEM) is one of the most important and ubiquitous numerical methods for solving partial differential equations (PDEs) on computers for scientific and engineering discovery. Applying the FEM to larger and more detailed scientific models has driven advances in high-performance computing for decades. Here we demonstrate that scalable spiking neuromorphic hardware can directly implement the FEM by constructing a spiking neural network that solves the large, sparse, linear systems of equations at the core of the FEM. We show that for the Poisson equation, a fundamental PDE in science and engineering, our neural circuit achieves meaningful levels of numerical accuracy and close to ideal scaling on modern, inherently parallel and energy-efficient neuromorphic hardware, specifically Intel’s Loihi 2 neuromorphic platform. We illustrate extensions to irregular mesh geometries in both two and three dimensions as well as other PDEs such as linear elasticity. Our spiking neural network is constructed from a recurrent network model of the brain’s motor cortex and, in contrast to black-box deep artificial neural network-based methods for PDEs, directly translates the well-understood and trusted mathematics of the FEM to a natively spiking neuromorphic algorithm.

Applied mathematics↗
DOE OSTI2024

A Low-Rank QTT-based Finite Element Method for Elasticity Problems

We present an efficient and robust numerical algorithm for solving the linear elasticity problem that combines the Quantized Tensor Train format and a domain partitioning strategy. This approach makes it possible to solve the linear elasticity problem on a computational domain that is more general than a square. By integrating Z-ordering and subdomain concatenation, our method substantially decreases memory usage and achieves a notable reduction in rank compared to established Finite Element implementations like the FEniCS platform. This efficiency is maintained while still guaranteeing exponential convergence with respect to the number of degrees of freedom. This performance gain, however, requires a fundamental rethinking of how core finite element operations are implemented. This includes changes to mesh discretization, node and degree of freedom ordering, stiffness matrix and internal nodal force assembly, and the execution of algebraic matrix-vector operations. In this work, we discuss all these aspects in detail and assess the method’s performance in the numerical approximation of three representative test cases.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Additional considerations in analytical solution for time-dependent heat conduction in a three-dimensional multilayer sphere

This work presents an analytical method to solve the heat conduction equation in three dimensions for problems consisting of multilayer concentric spheres. The method can be used to treat time-varying heat conduction problems where the heat source that drives the transient is time-invariant. Equally applicable to all Poisson-type problems with concentric spherical geometry, the method consists of representing the solution as a summation of weighted eigenfunctions. The weights for each eigenfunction are computed algebraically. Previous work has already established the core constituents of the methodology. The current work augments the existing methods by including consideration of nonzero interface resistance between layers and explicit discussion on the boundary condition homogenization required to treat inhomogeneous problems. Also, two demonstration problems are presented. One demonstration problem is based on the method of manufactured solutions and therefore allows for comparison with exact expressions for the solution temperature distribution. The second, more complex, demonstration problem relies on the finite element method for comparisons. The expected convergence behavior is observed for both demonstration problems.

97 - MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Application of physics-informed neural networks (PINNs) solution to coupled thermal and hydraulic processes in silty sands

Abstract The accurate modeling of water and heat transport in soils is crucial for both geo-environmental and geothermal engineering. Traditional modeling methods are problematic because they require well-defined boundaries and initial conditions. Recently, physics-informed neural networks (PINNs), which incorporate partial differential equations (PDEs) to solve forward and inverse problems, have attracted increasing attention in machine learning research. In this study, we applied PINNs to tackle hydraulic and thermal transport coupling forward problems in silty sands. A fully connected deep neural network was utilized for training. This neural network model leverages automatic differentiation to apply the governing equations as constraints, based on the mathematical approximations established by the neural network itself. We conducted forward problems and compared the solutions derived from PINNs with those from Finite Element Method (FEM) simulations. The forward problem results demonstrate the PINNs model’s capability in predicting hydraulic transport, heat transport, and thermal–hydraulic coupling in silty sands under various boundary conditions. The PINNs exhibited great performance in simulating the thermal–hydraulic coupling problem. The accuracy of the PINNs solutions shows its potential for simulation in geotechnical engineering.

Feng, Yuan↗
DOE OSTI2025

Boosting efficiency and reducing graph reliance: Basis adaptation integration in Bayesian multi-fidelity networks

The computational cost of high-fidelity numerical models makes outer-loop analysis, which requires repeated interrogation of the model such as uncertainty quantification, computationally demanding. Multi-fidelity methods, which construct a surrogate model using data from an ensemble of models of varying cost and accuracy, can substantially reduce the cost of outer-loop analysis. However, these methods can be difficult to apply when the model ensemble does not admit a clear hierarchy a priori and the correlations between models are low. Consequently, in this paper, we present a multi-fidelity method that leverages dimension reduction to enhance the correlation between models, thereby reducing the amount of data needed to train a surrogate from an unordered ensemble of models. Our method utilizes basis adaptation to build low-dimensional polynomial chaos expansions of each model and employs Multi-fidelity Networks to encode the relationships among models. We show that the resulting method exhibit two notable advantages over its counterpart: (1) enhanced accuracy (both reduced bias and variance); and (2) reduced dependency on the graph structure encoding relationships among models. We demonstrate the approach on an analytical test problem and a challenging finite element model for a spent nuclear fuel. Our method produces a surrogate model that is significantly more accurate than either a single-fidelity surrogate or a multi-fidelity surrogate constructed without basis adaptation.

42 ENGINEERING↗
DOE OSTI2025

Analysis of the weighted shifted boundary method for the Poisson and Stokes problems

The Shifted Boundary Method (SBM) belongs to the class of unfitted (or immersed, or embedded) finite element methods, and relies on reformulating the original boundary value problem over a surrogate (approximate) computational domain. Accuracy is maintained by properly shifting the location and values of the boundary conditions. This avoids integration over cut cells and the associated implementation issues. Recently, the Weighted SBM (WSBM) was proposed for the Navier-Stokes equations with free surfaces and the Stokes flow with moving boundaries. The attribute “weighted” in the name WSBM stems from the fact that its variational form is weighted with the elemental volume fraction of active fluid. The motivation for the development of the WSBM was the preservation of the volume of active fluid to a higher degree of accuracy, which in turn resulted in improved stability and robustness characteristics in moving-boundary, time-dependent simulations. In this article, we present the numerical analysis of the WSBM formulations for the Poisson and Stokes problems. We give mathematical conditions under which the bilinear forms defining the discrete variational formulations are uniformly coercive (Poisson problem) or inf-sup stable (Stokes problem). By these results, stability and optimal convergence is proven in the natural norm; L2-error estimates can also be derived.

Approximate domain boundaries↗
DOE OSTI2026

Input specific neural networks

Neural networks have emerged as powerful tools for mapping between inputs and outputs. However, their black-box nature limits the ability to encode or impose specific structural relationships between inputs and outputs. Many scientific and engineering problems, such as constitutive modeling in solid mechanics, require networks that can enforce convexity, monotonicity, or other structural constraints to ensure physical consistency. Here, we introduce the Input Specific Neural Network (ISNN), a new architecture that enables multiple, distinct constraints to be imposed on different input subsets for scalar-valued outputs. This framework unifies convex, monotone–convex, monotone, and arbitrary mappings within a single network for the first time. Two ISNN architectures with analytical first- and second-order derivatives are developed. We demonstrate the performance on synthetic toy problems, inverse problems in isotropic hyperelasticity, and finite element simulations. ISNNs achieve improved extrapolation behavior, require fewer invariant inputs than standard input convex networks for polyconvex potentials, and enable significant computational savings via manual differentiation. We also show how ISNNs can be used to learn structural relationships between inputs and outputs via a binary gating mechanism. Particularly, ISNNs are employed to model a homogenized anisotropic free energy potential in a decoupled multiscale setting. The network learns whether or not the potential should be modeled as polyconvex and retains only the relevant layers while using the minimum number of inputs. ISNNs provide a flexible foundation for embedding structural priors into neural networks, enhancing both interpretability and stability. They are broadly applicable across computational mechanics and other scientific domains requiring constrained functional relationships.

Jadoon, Asghar A. [Univ. of Texas, Austin, TX (Uni↗
DOE OSTI2025

Derivation and verification of the direct-sampling method for simulating Monte Carlo flight paths in tetrahedral meshes with linear finite-element cross sections

This paper provides a derivation of a direct-sampling approach for modeling continuously varying cross sections in tetrahedral-mesh-based Monte Carlo codes. Specifically, cross sections are spatially approximated using linear nodal finite elements. A linearization strategy is provided for non-linearly varying cross sections. The method is verified against seven analytical pure-absorber test problems. These test problems also highlight the benefit of using linear finite elements over element-wise-constant cross sections.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

Enriched immersed finite element and isogeometric analysis: algorithms and data structures

Immersed finite element methods provide a convenient analysis framework for problems involving geometrically complex domains, such as those found in topology optimization and microstructures for engineered materials. However, their implementation remains a major challenge due to, among other things, the need to apply nontrivial stabilization schemes and generate custom quadrature rules. This article introduces the robust and computationally efficient algorithms and data structures comprising an immersed finite element preprocessing framework. The input to the preprocessor consists of a background mesh and one or more geometries defined on its domain. The output is structured into groups of elements with custom quadrature rules formatted such that common finite element assembly routines may be used without or with only minimal modifications. The key to the preprocessing framework is the construction of material topology information, concurrently with the generation of a quadrature rule, which is then used to perform enrichment and generate stabilization rules. While the algorithmic framework applies to a wide range of immersed finite element methods using different types of meshes, integration, and stabilization schemes, the preprocessor is presented within the context of the extended isogeometric analysis. This method utilizes a structured B-spline mesh, a generalized Heaviside enrichment strategy considering the material layout within individual basis functions’ supports, and face-oriented ghost stabilization. Using a set of examples, the effectiveness of the enrichment and stabilization strategies is demonstrated alongside the preprocessor’s robustness in geometric edge cases. Additionally, the performance and parallel scalability of the implementation are evaluated.

Computer implementation↗
DOE OSTI2026

Feasibility Study on Implementing a Staggered-Grid Finite Volume Method for System Analysis Code Development Under the MOOSE Framework

Here, this work summarizes a feasibility study on testing numerical algorithms that are suitable and efficient for advanced system analysis code development under the mutli-physics framework, MOOSE. The key to the test bed is the implementation of high-order one-dimensional staggered-grid finite volume method (SG-FVM), and its direct interaction with the linear/nonlinear solver, PETSc. The test bed utilized a more flexible code structure to enable the finite volume method implementation and direct interacting with the solver package, instead of using the natively supported finite element method by the framework. Using a suite of selected test problems with different problem sizes and levels of complexity, the implemented SG-FVM demonstrated superior performance improvement against a direct finite element method implementation through MOOSE. On two computer systems, the speedup was observed to be significant, with at least one order of magnitude of solving time reduction. For a complex reactor model, transient simulation was performed using the newly developed finite volume method code, the results of which agree very well with the reference results from the finite element method code. Overall, this study demonstrates a successful feasibility study on the proposed numerical algorithms and software structure to support advanced system analysis tool development.

MOOSE↗
DOE OSTI2025

A note on the reliability of goal-oriented error estimates for Galerkin finite element methods with nonlinear functionals

Here, we consider estimating the discretization error in a nonlinear functional J (u) in the setting of an abstract variational problem: find u ϵ $\mathscr{V}$ such that B (u, φ) = L (φ) ∀φ ϵ $\mathscr{V}$, as approximated by a Galerkin finite element method. Here, $\mathscr{V}$ is a Hilbert space, B (. , .) is a bilinear form, and L (∙) is a linear functional. We consider well-known error estimates η of the form J (u) - J (u h ) ≈ η = L (z) - B (u h , z), where u h denotes a finite element approximation to u, and z denotes the solution to an auxiliary adjoint variational problem. We show that there exist nonlinear functionals for which error estimates of this form are not reliable, even in the presence of an exact adjoint solution z. An estimate η is said to be reliable if there exists a constant C ϵ $\mathbb{R}$ >0 independent of u h such that |J (u) - J (u h )| ≤ C|η|. We present several example pairs of bilinear forms and nonlinear functionals where reliability of η is not achieved.

A posteriori↗
DOE OSTI2025

Advanced System Thermal Fluids Solver Development for SAM

This work summarizes a feasibility study on testing numerical algorithms that are suitable and efficient for advanced system analysis code development under the mutli-physics framework, MOOSE. The key is the implementation of a high-order one-dimensional staggered-grid finite volume method (SG-FVM), and its direct interaction with the linear/nonlinear solver, PETSc. Leveraging the existing capabilities of the SAM code, significant code coverages were established in the finite volume method code. This in turn allows for a suite of test problems with different problem sizes and levels of complexity to be used to quantify the performance improvement of the finite volume method code. As evidently shown in this study, the implemented SG-FVM demonstrated superior performance improvement against a direct finite element method implementation through MOOSE for the wide range of selected problems. On two computer systems, the speedup was observed to be significant, with at least one order of magnitude of solving time reduction. In addition, for a complex reactor model, transient simulation was performed using the finite volume method code, the results of which agree very well with the reference results from the finite element method code. Overall, this study demonstrates a successful feasibility study on the proposed numerical algorithms and software structure to support advanced system analysis tool development. In this work, short-term priority development and testing items were identified, and long-term code adoption and integration plans were made for the eventual deployment of the finite volume method in the SAM code.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2025

Characterizing Artificial Viscosity Parameters with Approximate Symmetries

We are often faced with trying to capture the physics of compressible shocks, which are governed by the Euler equations. However, Euler shocks are formally discontinuous at the shock front (translating to a step-function behavior of rel evant flow variables). This poses a practical problem for codes with finite-sized grid elements. As a result, one must make a concession in simulating the be havior of shocks within a discrete framework. In particular, we must blur, or ‘regularize’ Euler shocks so that they may be captured on a finite grid.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Real-Time Bayesian Inference at Extreme Scale: A Digital Twin for Tsunami Early Warning Applied to the Cascadia Subduction Zone

We present a Bayesian inversion-based digital twin that employs acoustic pressure data from seafloor sensors, along with 3D coupled acoustic–gravity wave equations, to infer earthquake-induced spatiotemporal seafloor motion in real time and forecast tsunami propagation toward coastlines for early warning with quantified uncertainties. Our target is the Cascadia subduction zone, with one billion parameters. Computing the posterior mean alone would require 50 years on a 512 GPU machine. Instead, exploiting the shift invariance of the parameter-to-observable map and devising novel parallel algorithms, we induce a fast offline–online decomposition. The offline component requires just one adjoint wave propagation per sensor; using MFEM, we scale this part of the computation to the full El Capitan system (43,520 GPUs) with 92% weak parallel efficiency. Moreover, given real-time data, the online component exactly solves the Bayesian inverse and forecasting problems in 0.2 seconds on a modest GPU system, a ten-billion-fold speedup.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

A Domain-Decomposed A-ϕ Formulation Based on Lagrange Multipliers for Low-Frequency Problems

A domain-decomposed A-ϕ formulation based on Lagrange multipliers is proposed to simulate low-frequency elec- tromagnetic problems. This method partitions the computational domain into smaller subdomains, allowing each subdomain to be independently formulated using Lagrange multipliers as Dirichlet boundary conditions, while ensuring continuity of the fields across the interfaces. A mixed finite element method, utilizing both vector and scalar basis functions, is employed to discretize the formulation, resulting in a global system to be solved. The proposed method is validated using TEAM Problem 7 at 50 Hz, demonstrating its effectiveness in handling complex geometries and addressing the low-frequency breakdown issues commonly encountered in traditional finite element methods.

Hossain, Amzad↗